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arXiv:1407.4883 · Physical Review A

Comparing resolved-sideband cooling and measurement-based feedback cooling on an equal footing: Analytical results in the regime of ground-state cooling

Kurt A. Jacobs, Hendra I. Nurdin, Frederick W. Strauch, M. R. James

Published Apr 8, 2015• 25 citations
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Abstract

We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling---an example of coherent feedback control---and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing---that is, for the same interaction strength---and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.

DOI
10.1103/physreva.91.043812

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We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling---an example of coherent feedback control---and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing---that is, for the same interaction strength---and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.
We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling --- an example of coherent feedback control --- and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing --- that is, for the same interaction strength --- and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength.
Coherent feedback control, which does not destroy the coherence of quantum signals by measurement, outperforms measurement-based feedback control in quantum systems. Nevertheless, systematic methods for designing coherent feedback controllers are absent so far. Thus, expectations for controller design methods are growing from control engineering perspectives. This paper proposes a method for designing a coherent feedback controller based on frequency analysis. Previous studies have rarely considered frequency analysisbased approaches. The paper theoretically reveals that groundstate cooling in the unresolved sideband regime is possible using a simple feedback structure with only a single double-sided cavity as the controller. While the proposed design method is a natural extension of the method based on classical control theory, it has rarely been used in coherent feedback control, thus offering new possibilities for coherent controller design methods.
We introduce a general framework, based on collision models and discrete completely positive maps, to describe on an equal footing coherent and measurement-based feedback control of quantum mechanical systems. We apply our framework to prominent tasks in quantum control, ranging from cooling to Hamiltonian control. Unlike other proposed comparisons, where coherent feedback always proves superior, we find that either measurements or coherent manipulations of the controller can be advantageous depending on the task at hand. Measurement-based feedback is typically superior in cooling, while coherent feedback is better at assisting quantum operations. Furthermore, we show that both coherent and measurement-based feedback loops allow one to simulate arbitrary Hamiltonian evolutions, and discuss their respective effectiveness in this regard.
We revisit the problem of preparing a mechanical oscillator in the vicinity of its quantum-mechanical ground state by means of feedback cooling based on continuous optical detection of the oscillator position. In the parameter regime relevant to ground state cooling, the optical back-action and imprecision noise set the bottleneck of achievable cooling and must be carefully balanced. This can be achieved by adapting the phase of the local oscillator in the homodyne detection realizing a so-called variational measurement. The trade-off between accurate position measurement and minimal disturbance can be understood in terms of Heisenberg's microscope and becomes particularly relevant when the measurement and feedback processes happen to be fast within the quantum coherence time of the system to be cooled. This corresponds to the regime of large quantum cooperativity Ctextqgtrsim1C_{\\text{q}}\\gtrsim1Ctextq​gtrsim1, which was achieved in recent experiments on feedback cooling. Our method provides a simple path to further pushing the limits of current state-of-the-art experiments in quantum optomechanics.
Linear--quadratic--Gaussian (LQG) control is optimal only with respect to a prescribed cost function, the choice of which dictates the physical objective of the control. We consider feedback cooling of a continuously monitored quantum harmonic oscillator by shifting the minimum of its trapping potential. In this setting, the physically relevant cooling objective can be defined as minimizing the oscillator's energy relative to the feedback-shifted potential. In contrast, conventional LQG control evaluates the energy from a fixed origin and thus fails to directly optimize this quantity. To address this problem, we introduce a redesigned cost function that explicitly accounts for the feedback-induced shift of the potential. We then derive the corresponding optimal feedback law and obtain an analytic expression for the minimum achievable steady-state phonon occupation number. The redesigned LQG control achieves a lower occupation number than low-pass-filter (LPF) feedback formulated for the same cooling objective. While this improvement is minor at detection efficiencies currently attainable in experiments---indicating that LPF feedback already delivers near-optimal cooling performance---the advantage becomes pronounced as the detection efficiency approaches unity. In this regime, the redesigned LQG control provides an increasing advantage for reaching the motional ground state at a finite measurement strength. We clarify that the conventional and redesigned cost functions represent distinct control objectives rather than different implementations of the same optimization problem.
We study the cooling performance of optical-feedback controllers for open optical and mechanical resonators in the linear quadratic Gaussian setting of stochastic control theory. We utilize analysis and numerical optimization of closed-loop models based on quantum stochastic differential equations to show that coherent control schemes, where we embed the resonator in an interferometer to achieve all-optical feedback, can outperform optimal measurement-based feedback control schemes in the quantum regime of low steady-state excitation number. These performance gains are attributed to the coherent controller's ability to simultaneously process both quadratures of an optical probe field without measurement or loss of fidelity, and may guide the design of coherent feedback schemes for more general problems of robust nonlinear and robust control.